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ISSN 2229 – 5046SIMILARITY SOLUTIONS OF SPHERICAL SHOCK WAVES
IN AN IDEAL GAS WITH THERMAL RADIATION
ANOOP KUMAR
*, RAJAN ARORA
Department of Applied Science and Engineering,
IIT Roorkee, Saharanpur Campus, Saharanpur – (U.P.)-247001, India.
Received On: 25-05-17; Revised & Accepted On: 16-06-17)
ABSTRACT
I
n this paper, a group theoretic method is used to obtain an entire class of similarity solutions to the problem of shocks propagating through in an ideal gas with thermal radiation, and to characterize analytically the state dependent form of the medium ahead for which the problem is invariant and admits similarity solutions. The arbitrary constants occurring in the expressions for the infinitesimals of the local Lie group of transformations give rise to two different cases of possible solutions i.e. with a power law and exponential shock paths. A particular case of collapse of imploding spherically symmetric shock in a medium in which the initial density obeys power law is worked out in detail. Numerical calculations have been performed to obtain the similarity exponents and the profiles of the flow variables behind the shock, and comparison is made with the Guderley’s [1] results.Keywords: Lie group, Similarity solutions, Gas-dynamics, Shock Waves, Rankine-Hugoniot Conditions.
1. INTRODUCTION
Nonlinear partial differential equations (NPDEs) are widely used to describe complex phenomena in various fields of physical and engineering interests. Many flow fields involving wave phenomena are governed by quasi linear hyperbolic system of nonlinear partial differential equations (PDEs). For nonlinear systems involving discontinuities such as shocks, we do not generally have the complete exact solutions, and we have to rely on some approximate analytical or numerical methods which may be useful to provide information to understand the physics involved therein. One of the most powerful methods to obtain the similarity solutions to such type of nonlinear PDEs is the similarity method which is based upon the study of their invariance with respect to one parameter Lie group of transformations. Indeed, with the help of symmetry generators, one can construct similarity variable which can reduce the system of partial differential equations (PDEs) to a system of ordinary differential equations (ODEs). A theoretical study of the imploding shock wave near the centre of convergence in an ideal gas was first performed by Guderley [1]. Among the extensive work that followed, we mention the contributions of Sakurai [2], Zeldovich and Raizer [3], Hayes [4], Axford and Holm [5, 6], Lazarus [7], Hafner [8], Sharma and Radha [9], Jena and Sharma [10], Conforto [11], Madhumita and Sharma [12], Sharma and Radha [13], Sharma and Arora [14], Arora et al. [15, 16] and Husain et al. [17] who presented high accuracy results and alternative approaches for the investigation of implosion problem.
Steeb [18] determined the similarity solutions of the Euler equations and the Navier–Stokes equations for incompressible flows using the group theoretic approach outlined in the work of Bluman and Cole [19], Ovasiannikov [20], Olver [21], Logan [22] and Bluman and Kumei [23]. In the present paper, following Bluman and Kumei [23], and in a spirit closer to Logan [22], we characterize the medium ahead of shock for which the problem is invariant and admits similarity solutions. The occurrence of arbitrary constants in the expressions for the infinitesimals of the Lie group of transformations yields different cases of solutions with a power law and exponential shock paths. We have worked out in detail one particular case of collapse of imploding spherically symmetric shock in a medium in which initial density obeys power law.
Corresponding Author: Anoop Kumar
*,
Department of Applied Science and Engineering,
© 2017, IJMA. All Rights Reserved 31
The type of motion in which the distribution of flow variables remain similar to themselves with time and vary only as a result of changes in scale is called self-similar. For self-similar motions, the system of non-linear partial differential equations reduces to a system of ordinary differential equations in new unknown functions of the similarity variable
ξ
.
Here, we consider the spherically symmetric motion of a polytropic gas with adiabatic indexγ
and use the Lie group of transformations to establish the self-similar solutions. The arbitrary constants occurring in the expressions for the infinitesimals of the local Lie group of transformations give rise to two different cases of possible solutions i.e. with a power law and exponential shock paths. The computed values of the similarity exponent(
δ
)
are also compared with the Guderley [1] result and the computation of the flow field in the region behind the shock has been carried out to determine the effects of the ambient density exponent(
θ
).
2. BASIC EQUATIONS AND SHOCK CONDITIONS
The basic equations [24] describing the one-dimensional unsteady spherically symmetric motion in an ideal gas can be written as
t t
t
2
/
0,
1
0,
2
(
1)
0,
x x
x x
x x
u
u
u x
u
u u
p
u
p
u p
p u
q
x
ρ
ρ
ρ
ρ
ρ
γ
γ
ρ
+
+
+
=
+
+
=
+
+
+
+
−
=
(1)
where
p
is the gas pressure,ρ
is the density,u
is the velocity,γ
is the constant specific heats ratio, the independent variablex
is the radial distance from the center in spherically symmetric flows;t
is the time;q
is the cooling rate. Here, we assume thatq
is given as:,
)
,
(
ρ
T
q
0ρ
αT
βq
=
(2)where
q
0,α
andβ
are constants and q is defined as the total amount of heat energy liberated per unit mass per unit time over the whole frequency interval. The non-numeric subscripts denote the partial differentiation with respect to the indicated variables unless stated otherwise. The equation of state for thermally perfect gas is given in the following form:,
T
R
p
=
ρ
(3)where T is the translational temperature and R is the specific gas constant. Now, we consider the shock speed
dt
dX
v
=
/
propagating into an inhomogeneous medium specified by,
,
0
p
p
0u
=
=
ρ
=
ρ
0(
x
)
. (4)The Rankine-Hugoniot jump conditions for the strong shocks are (Singh and Vishwakarma [24])
2
0 0
2
1
2
,
( ( )),
( ( ))
,
1
1
1
u
v
ρ
γ
ρ
X t
p
ρ
X t v
γ
γ
γ
+
=
=
=
+
−
+
(5)where the suffix 0 refers to the condition of flow just ahead of the shock.
3. SIMILARITY ANALYSIS BY INVARIANCE GROUPS
Here, we suppose that there exists a solution of system (1) along a family of curves, called similarity curves for which the system (1) of partial differential equations reduces to a system of ordinary differential equations; this type of solution is called a similarity solution. In order to obtain the similarity solutions of the system (1), we derive its symmetry group such that the system is invariant under this group of transformations. The idea of the calculation is to find a one-parameter infinitesimal group of transformations (Sharma and Arora [14]):
),
,
,
,
,
(
),
,
,
,
,
(
),
,
,
,
,
(
),
,
,
,
,
(
),
,
,
,
,
(
*
* *
* *
p
u
t
x
P
p
p
p
u
t
x
S
p
u
t
x
U
u
u
p
u
t
x
t
t
p
u
t
x
x
x
ρ
ε
ρ
ε
ρ
ρ
ρ
ε
ρ
ψ
ε
ρ
χ
ε
+
=
+
=
+
=
+
=
+
=
(6)
In continuation, we shall use summation convention, and introduce the notation
p
u
u
u
u
x
x
t
x
1=
,
2=
,
1=
ρ
,
2=
,
3=
and,
j i i
j
x
u
p
∂
∂
=
where i=1, 2, 3 and j=1, 2.The system of basic equations (1) which is represented as
,
3
,
2
,
1
,
0
)
,
(
x
,u
p
=
k
=
F
k j i ijis said to be constantly conformally invariant under the infinitesimal group of transformations (6), if there exist constants
α
kr(
k
,
r
=
1
,
2
,
3
)
such that for all smooth surfaces,u
i=
u
i(
x
j)
, we have
L
F
k=
α
krF
r,
(7)
,
11 r
F
rF
L
=
α
⇒
=
α
11F
1+
α
12F
2+
α
13F
3 etc.,where
L
is the Lie derivative in the direction of the extended vector field:,
L
ij i j i i
j j
p
u
x
∂
∂
+
∂
∂
+
∂
∂
=
ξ
η
β
with
ξ
1=
ψ
,
ξ
2=
χ
,
η
1=
S
,
η
2=
U
,
η
3=
P
,
and,
n j i l n l i l j l k j k i
j i i
j
p
p
u
p
x
p
u
x
∂
∂
−
∂
∂
−
∂
∂
+
∂
∂
=
η
η
ξ
ξ
β
(8)
where
l
=
1
,
2
,
j
=
1
,
2
,
i
=
1
,
2
,
3
,
n
=
1
,
2
,
3
andk
=
1
,
2
,
3
.
Here, repeated indices imply summation convention andβ
ij is the generalization of the derivative transformation.Therefore, equation (7) can be written as:
.
3
,
2
,
1
,
3
,
2
,
1
,
=
=
=
∂
∂
+
∂
∂
+
∂
∂
r
k
F
p
F
u
F
x
F
r r k i j k i j i k i
j k
j
η
β
α
ξ
(9)Substitution of
β
ij from (8) into (9) yields a polynomial equation inp
ij.
Setting the coefficients ofi j
p
andp
ijp
nj to zero, we obtain a system of first order linear partial differential equations in the generators ψ, χ, S, U and P. This system, which is called the system of the determining equations, is given by11 13 13
11 12 13
11
11 12 13 12 13
2
2
2
(
1)
,
,
,
,
,
,
1
,
t x x
t x u x p
t x
x u u
p p
u
u
p u
S
u S
U
U
u S
q
x
x
x
x
S
u
S
S
U
u S
u
U
u
S
u S
U
u
p
uS
U
u
ρ
ρ ρ
ρ χ
ρ
γ
ρ
ρ
α
α
α γ
ρ
ψ
ψ
α
ρψ
α
α
χ
χ
ρ
α
ρχ
ρ
α ρ α
α γ
ρ
α
α
ρ
+
+
+
+
−
=
+
+
−
−
−
=
−
=
=
−
+
−
+
=
−
+
+
=
+
+
+
=
+
(10)
21 23 23
21 22
21 23
21 22 23 22 23
2
1
2
2
(
1)
,
,
,
1
1
,
,
1
,
1
1
1
,
t x x
u t x
p x
t u x u
p p x
u
p u
U
uU
P
q
x
x
U
U
u
uU
P
u
U
U
uU
u
P
u
p
S
uU
P
u
ρ
ρ ρ
ρ
γ
α
α
α γ
ρ
ρ
α
ψ
ψ
α
α
ψ
α
ρ
ρ
χ
χ
α ρ α
α γ
ρ
χ
α
α
ρ
ρ
ρ
ρ
+
+
=
+
+
−
=
−
−
=
+
=
−
=
−
+
−
+
=
+
+
−
+
+
−
=
+
© 2017, IJMA. All Rights Reserved 33
31 33
31 32 33
31
31 32 33 32 33
2
(
1)
(
1)
2
2
(
1)
,
,
,
,
,
,
1
.
t x x p
u x p t x
u x u
t p p x
p u
P
u P
p U
u P
pU
S q
S q
P q
x
x
u
pu
q
x
x
P
P
p
P
u
u P
p U
u
P
pU
p
u P
u
p
U
pU
u P
u
u
ρ
ρ
ρ ρ
γ
γ
χ
γ
γ
ρ
ρ
γ
α
α
γ
ρ
α
γ ψ
α
ψ
ψ
α
γ
α
γ
γ
γ χ
α ρ α
α γ
χ γ
χ
α
α
ρ
+
+
+
+
−
+ −
+ −
+
=
+
+ −
=
−
=
−
−
=
+
=
+
−
+
=
+
+
−
+
+
−
=
+
(12)
Solving the above systems of determining equations, we obtain the infinitesimals of the group of transformations as follows:
11 22
22 11 22
22
(
) ,
(
) ,
(2
3 ) ,
(
2 ) ,
,
(2
),
p
S
a
U
a u
P
a p
a x
at
b
S q
ρP q
q
a
α
ρ
α
α
α
χ
α
ψ
α
=
+
=
+
=
+
+
=
+
=
+
+
=
+
(13)
where
a
,
b
,
α
11 andα
22 are the arbitrary constants. The arbitrary constants, occurring in the expressions of the infinitesimals of local Lie-group of transformations, give rise to four different cases of possible solutions.4. SELF-SIMILAR SOLUTIONS AND CONSTRAINTS
The arbitrary constants, which become observable in the expressions for the infinitesimals of the invariant group of transformations, yield four different cases of possible solutions as discussed below:
Case-I: When
a
≠
0
andα
22+
2
a
≠
0
,
the change of variables from (x, t) to( )
x
,
t
defined as,
a
b
t
t
x
x
=
=
+
(14)does not change the equation (1)-(12). Thus, rewriting the set of equations (13) in the expressions of the new variables
x
andt
,
and then suppressing the bar sign, we obtain11 22
22 11 22
22
(
) ,
(
) ,
(2
3 ) ,
(
2 ) ,
,
(2
).
p
S
a
U
a u
P
a p
a x
at
S q
ρP q
q
a
α
ρ
α
α
α
χ
α
ψ
α
=
+
=
+
=
+
+
=
+
=
+
=
+
(15)
The similarity variable and the forms of similarity solutions for
ρ
,
u
,
p
andq
follow from the invariant surface conditions which yield22
,
,
,
(2
).
t x t x
t x p
S
u
u
U
p
p
P
S q
ρPq
q
a
ψ ρ χ ρ
ψ
χ
ψ
χ
α
+
=
+
=
The equations (16) with the help of (15) yield on integration the forms of the flow variables as
( )
( )
[ ]
11
11 1
1
2 ( 1)
2 3 (2 1)
( ) ,
( ) ,
( ) ,
( ) ,
a
k
a a
t
S
u
t
U
p
t
P
q
p
Q
α
δ
δ
δ δ α
ρ
ξ
ξ
ξ
η
+
−
− +
− − +
=
=
=
=
(17)along with the similarity curve
,
δ
ξ
t
x
=
(18) where Q is an arbitrary function ofη
which is found as( )
( )
,
11 11
1 2
+ −
+ −
=
δ αα
ρ
η
aa
p
a
a
k
=
α
11+
and
a
a
2
22
+
=
α
δ
(19)and
S
,
U
andP
are the arbitrary functions ofξ
.
Since the shock is a similarity curve, and at shock ξ remains constant; without loss of generality it may be normalized to be atξ
=
1
.
The shock pathX
=
X
(
t
),
shock velocityv
and the values of the density, velocity and pressure of gas atξ
=
1
are then given by,
δ
t
X
=
(20).
1
−
=
=
δ
δ
δt
t
X
v
(21)At the shock, we have the following conditions on the functions
S
,
U
andP
( )
11
1 1
1 1
2 ( 1) 1
(1) ,
(1) ,
(1).
a
k
t
S
u
t
U
p
t
P
α
ξ δ ξ
δ ξ
ρ
+ =− =
− + =
=
=
=
(22)
The invariance of jump condition (5) for
ρ
suggests the following form ofρ
0(
x
)
:
,
)
(
0
θ
ρ
ρ
x
=
cx
(23)where
ρ
c is an arbitrary constant and 11.
a
a
δ
α
θ
=
+
(24)Also, we obtain the following conditions on the functions
S
,
U
andP
at the shock(
ξ
=
1
)
:
2
1
2
(1)
,
(1)
,
1
1
2
(1)
,
(
1)
c
c
S
U
P
γ
ρ
δ
γ
γ
ρ δ
γ
+
=
=
−
+
=
+
(25)where
ρ
c is some reference constant associated with the medium. Using (20), (21) and (23) we rewrite the equation (17) as* *
0
2 * 0
(
( ))
( )
( ),
(
( ))
( ),
X t
S
u
v U
p
X t
v P
ρ ρ
ξ
ξ
ρ
ξ
=
=
=
(26)where
c c
P
P
U
U
S
S
ρ
δ
ξ
ξ
δ
ξ
ξ
ρ
ξ
ξ
* * 2*
(
)
)
(
,
)
(
)
(
,
)
(
)
(
=
=
© 2017, IJMA. All Rights Reserved 35
Substituting (26) in the equations of system (1) and using (18), (19), (20), (21) and (23), we obtain the following system of ordinary differential equations in
S
*,
U
*andP
*which on suppressing the asterisk sign becomes' '
' 1 '
' 1 '
*
2
(
)
0,
(
)
(
1)
0,
2
(
)
2 (
1)
(
1)
0,
S U
U
S
S
SU
U
U S
U S
P
U
U
P
P
P
P U
S q
ξ
θ
ξ
ξ
δ
δ
ξ
δ
δ
θ γ
γ
ξ
−−
−
+
+
+
=
−
+
−
+
=
−
+
−
+
+
+
+
−
=
(27)
where prime denotes the differentiation with respect to
ξ
and * 0β
δ
(2β−3)ρ
α β (α−β).
=
P
S
R
q
q
cThe jump conditions for the strong shock are:
2
1
2
(1)
,
(1)
,
(1)
.
1
1
1
U
S
γ
P
γ
γ
γ
+
=
=
=
+
−
+
(28)Case-II: When
a
=
0
andα
22+
2
a
≠
0
orα
22≠
0
,
the similarity variable and the forms of similarity solutions for the flow variables readily follow from (13) and (16), and can be expressed in the following forms on suppressing the bar signs),
(
),
(
))
(
(
* *0
ξ
ξ
ρ
ρ
=
X
t
S
u
=
v
U
),
(
))
(
(
2 *0
ξ
ρ
X
t
v
P
p
=
(
)
,
11
2 2
η
α δδ
Q
p
q
b
+
=
(29)Where *
(
)
,
*(
)
,
* 2(
)
,
c c
P
P
U
U
S
S
ρ
δ
ξ
δ
ξ
ρ
ξ
=
=
=
11,
11
2
+ −
=
δ αα
ρ
η
bp
,
0
2
)
2
(
)
(
2
θ
α
−
β
+
δ
+
δθ
β
−
δ
=
and the similarity variable
ξ
, the shock locationX
=
X
(
t
)
, the shock velocityv
andρ
0 are given by,
,
,
t tt
e
v
e
X
xe
δ δδ
δξ
=
−=
=
0
(
)
,
θ
ρ
ρ
x
=
cx
(30)with
,
,
11.
22 11
22
α
δθ
α
α
θ
α
δ
=
=
=
=
b
k
b
It may be noticed that this case leads to a class of similarity solutionswith an exponential shock path given in the second equation in (30). Substituting (29) into the equations of system (1) and using (30), we obtain the following system of ordinary differential equations in
S
*,
U
*,
andP
*which on suppressing the asterisk signs become:' '
' '
' '
*
2
(
)
0,
(
)
0,
2
(
)
(2
)
(
1)
0,
SU
U
S
S
SU
U
U S US
P
U
U
P
P
P U
S q
ξ
θ
ξ
ξ
ξ
θ
γ
γ
ξ
−
+
+
+
=
−
+
+
=
−
+
+ +
+
+
−
=
(31)
where prime denotes the differentiation with respect to
ξ
and * 0 (2 3) ( ).
β α β α β β
δ
ρ
− −
=
P
S
R
q
q
cThe jump conditions are:
2
1
2
(1)
,
(1)
,
(1)
.
1
1
1
U
S
γ
P
γ
γ
γ
+
=
=
=
+
−
+
(32)
The system (31) is to be solved subject to the jump conditions (32) for a shock of infinite strength.
Case-IV: When
a
=
0
andα
22−
α
11=
0
, the situation is similar to the case III in the sense that it does not permit for the existence of similarity solutions in spherically symmetric flow.5. IMPLODING SHOCKS
Now, we consider in detail the Case I of an imploding shock for which
v
>>
a
0,
wherea
0 is the speed of sound. For the problem of a converging shock collapsing at the center, the origin of timet
is taken to be the instant at which the shock reaches the center so thatt
≤
0
in equation (27). In this regard, the definition of the similarity variable is a little modified by setting,
)
(
,
)
(
t
δξ
x
t
δX
=
−
=
−
(33)so that the intervals of the variables are
−
∞
<
t
≤
0
,
X
≤
x
<
∞
and1
≤
ξ
<
∞
.
At the instant of collapse),
0
(
t
=
the gas velocity, pressure, density and the sound speed at any finite radius x are bounded, but witht
=
0
and finite x,ξ
=
∞
.
In order for the quantitiesu
,
p
,
ρ
and a to be bounded whent
=
0
and x is finite, we have the following boundary conditions atξ
=
∞
.
.
0
)
(
)
(
,
0
)
(
=
∞
∞
=
∞
S
P
U
(34)In the matrix notation system (27) can be written as
,
'
B
W
A
=
(35)where
W
=
(
U
,
S
,
P
)
tr,
and the matrixA
and the column vectorB
can be examined by inspection of system (27). In system (27) there is an unknown parameterδ
,
which cannot be obtained from an energy balance or the dimensional considerations; it is computed only by solving a non-linear eigenvalue problem for a system of ordinary differential equations. The range of similarity variable is1
≤
ξ
<
∞
for the implosion problem, and system (35) can be solved for the derivativesU
',
S
' andP
'in the following form:,
,
,
'
1 ' 2 ' 3∆
∆
=
∆
∆
=
∆
∆
=
S
P
U
(36)where
∆
, defined as the determinant of the matrix A, is given by(
) (
)
2,
−
−
−
−
=
∆
S
P
U
S
U
ξ
ξ
γ
(37)and
∆
k(
k
=
1
,
2
,
3
)
are the determinants obtained from∆
by replacing the kth column by the column vector,
B
and are given by(
)
(
1
)
(
)
2
(
1
)
2
(
1
)
*,
1
−
−
−
+
+
−
+
+
−
−
−
=
∆
U
U
S
U
P
P
PU
γ
Sq
ξ
γ
δ
δ
θ
ξ
δ
δ
ξ
(
)
(
)
2 *
2
(
1)
2 (
1)
2
(
1)
2
,
U S
P
PU
S
U
P
Sq
US
S
U
S
P
δ
ξ
θ
δ
γ
γ
δ
δ
ξ
θ
ξ
γ
ξ
− −
−
∆ =
−
+
+
+
+ −
+
+
−
−
(
) (
)
2
(
1
)
2
(
1
)
*(
1
)
.
3
−
+
−
−
−
−
−
−
−
−
−
=
∆
U
U
S
P
P
PU
Sq
U
S
γ
P
δ
δ
γ
ξ
γ
δ
δ
θ
ξ
ξ
It can be verified that
∆
is negative atξ
=
1
and positive atξ
=
∞
representing thereby that there exists a[
∞
)
∈
1
,
ξ
at which∆
vanishes, and accordingly the solutions become singular. In order to get a non-singular© 2017, IJMA. All Rights Reserved 37
1
∆
is zero too. It can be checked that at points where∆
and∆
1 vanish, the determinants∆
2 and∆
3 also vanish at the same time. To find the value of exponentδ
in such a manner, we introduce the variable Z as,
)
(
)
(
)
)
(
(
)
(
2ξ
ξ
γ
ξ
ξ
ξ
S
P
U
Z
=
−
−
(38)
whose derivative, in view of (36), is
.
/
)
)(
(
2
3 2 21
'
∆
∆
+
∆
−
∆
−
∆
−
=
S
P
S
U
Z
ξ
γ
γ
(39)
Equations (36), in view of (39), become
,
,
,
4 3 4
2 4
1
∆
∆
=
∆
∆
=
∆
∆
=
dZ
dP
dZ
dS
dZ
dU
(40)
where
2
(
)(
)
2,
2 3 1
4
+
∆
∆
−
∆
−
∆
−
=
∆
S
P
S
U
ξ
γ
γ
with
.
)
(
)
(
)
(
)
(
2 1
+
+
=
ξ
ξ
γ
ξ
ξ
ξ
S
P
Z
U
6. GUDERLEY SOLUTION
For a converging shock wave, we use the variable
ξ
defined earlier as,
ξ
δ
t
x
=
(41)where
−
∞
<
t
≤
0
and1
≤
ξ
<
∞
; here we useξ
as an independent variable in place ofx
. In terms of this independent variableξ
, let the density, velocity and pressure be given by:2
0
( ),
( ),
0( ).
x
x
G
u
V
p
P
t
t
δ
δ
ρ ρ
=
ξ
=
ξ
=
ρ
ξ
(42)Under the transformation (42), the given system (1) is transformed into following system:
'
' ' '
' '
*
(
1)
(
1) 3 ,
(2
)
1
(
1)
,
.
1
(
1)
3
(
1) 2
(
1)
G
V
V
V
V
G
P
P
V V
V V
G
G
P V
P V
PV
P V
P V
G q
ξ
ξ
θ
ξ
θ
ξ
δ
ξ
γ ξ
γ
θ
γ
δ
− +
= −
− −
+
− +
= −
−
−
− +
= −
−
− −
−
− −
(43)
And the shock conditions at the strong shock front are given by
(
1)
2
2
(1)
,
(1)
,
(1)
.
(
1)
(
1)
(
1)
G
γ
V
P
γ
γ
γ
+
=
=
=
−
+
+
(44)In the matrix notation the system (43) can be written as
,
'
B
W
A
=
(45) whereW
=
(
V
,
G
,
P
)
trand the matrix A and the column vector B can be identified by inspection of system (43). The unknown parameterδ
appearing in system (43) is computed only by solving a non-linear eigenvalue problem for a system of ordinary differential equations. System (43) can be solved for the derivativesV
',
G
'andP
' in the following form:' ' 3
1 2
'
,
,
,
V
=
∆
G
=
∆
P
=
∆
where
∆
, which is the determinant of the matrix A, is given by(
1
)
,
)
1
(
2 3
−
−
−
−
=
∆
G
P
V
G
V
ξ
γ
(47)
and
∆
k(
k
=
1
,
2
,
3
)
are the determinants obtained from∆
by replacing the kth column by the column vector B, and are given by(
)
,
)
1
(
)
1
(
2
)
1
(
3
1
)
2
(
)
1
(
)
1
(
* 1
+
−
+
−
+
−
+
−
−
−
−
+
−
−
=
∆
q
G
V
P
V
P
V
P
G
V
G
P
V
V
G
V
γ
δ
δ
θ
γ
ξ
ξ
θ
δ
ξ
(
)
(
)
(
)
2 * *1
(
)
(2
)
1
1
3
(
1) 2 (
) (
1)
1
1
3
(
1) 2 (
) (
1)
(
1) 3
,
1
(
)
(2
)
P
V V
V
G
PV
P V
P V
G q
G
V
PV
P V
P V
G q
V
V
P
P
V V
G
θ ξ
δ
ξ
ξ γ
θ
γ
δ
ξ
γ
θ
γ
δ
θ
γ ξ
θ
δ
−
−
−
+
− +
∆ =
+
− +
−
+ −
−
−
−
− −
−
− −
+
+
− +
−
+
+
(
)(
)
.
)
2
(
)
1
(
)
1
(
)
1
(
2
)
1
(
3
1
)
1
(
* 3
−
+
+
+
−
−
−
−
−
−
−
−
−
−
=
∆
θ
ξ
γ
γ
θ
γ
ξ
ξ
G
P
V
V
P
q
G
V
P
V
P
V
P
V
G
V
We introduce the variable Z as
(
)
,
)
(
)
(
1
)
(
)
(
2
−
−
=
ξ
ξ
γ
ξ
ξ
G
P
V
Z
(48)whose derivative, in view of (46), is
.
/
)
1
(
2
1 3 2 2'
∆
−
∆
−
∆
+
∆
=
G
P
G
V
Z
γ
γ
(49)
Equations (46), in view of (49), become
,
,
,
4 3 4 2 4 1∆
∆
=
∆
∆
=
∆
∆
=
dZ
dP
dZ
dG
dZ
dV
(50)where
∆
4=
2
(
−
1
)
∆
1−
∆
3+
2∆
2
.
G
P
G
V
γ
γ
7. NUMERICAL RESULTS AND DISCUSSION
© 2017, IJMA. All Rights Reserved 39
Table-1: Similarity exponent
δ
for spherically symmetric flow and the ambient density exponentθ
with,
66
.
1
=
γ
q0 =1, R=8.314,α
=0.5,ρ
c =1 andβ
=
1
.
5
.
0.0 0.2 0.4 0.6 0.8 1.01
0.0 0.2 0.4 0.6 0.8 1.0 Velocity
0.0 0.2 0.4 0.6 0.8 1.0
1
010 20 30 40 50 Density
(a) (b)
0.0
0.2
0.4
0.6
0.8
1.0
1
0.0
0.5
1.0
1.5
2.0
Pressure
(c)
Figure-1: The profiles of velocity, density and pressure in (a), (b) and (c), respectively, behind spherical shock for
θ
=
0
.
1
,
0
.
5
and1
.
3
.
RESULTS AND CONCLUSION
In the present work, we have used the method of Lie group of transformations to analyze the existence of self-similar solutions for a hyperbolic system of non-linear partial differential equations governing the problem of spherical shock waves in an ideal gas with thermal radiation. The importance of using the Lie group of transformations for obtaining similarity solutions is due to the fact that the arbitrary constants occurring in the expressions for the generators of the local Lie-group of transformations give rise to different cases of possible solutions. The form of the similarity variables and similarity solutions are suggested by the method itself after suitably considering the arbitrary constants. It was observed that, depending upon arbitrary constants appearing in the infinitesimals of the transformations, we obtain different solutions with power law and exponential shock paths. In both the cases, the corresponding reduced systems of ordinary differential equations were found out. The similarity exponent
δ
and the flow variables can only be obtained after solving the system of ordinary differential equations numerically. We integrated the equations (40)
θ
Computed(
δ
)
Guderley’s [1](
δ
)
% Error 0.10.5 1.0 1.3 1.6 2.0
0.674677 0.635475 0.590995 0.570308 0.548998 0.551405
0.685196 0.622135 0.590085 0.570305 0.548998 0.525370
1.00% 2.14% 0.15% 0.00% 0.00% 4.72%
3
.
1
=
θ
θ
=
0
.
1
3
.
1
=
θ
5
.
0
=
θ
1
.
0
=
θ
1
.
0
=
numerically using the fourth order Runge-Kutta method for
1
≤
ξ
<
∞
and the values of velocity, density and pressure behind the shock are plotted in Figures 1 (a), (b) and (c), respectively. Figures 1 (a), (b) and (c) show that behind the shock the velocity decreases and the density increases monotonically as we move towards the center of collapse whereξ
→
∞
,
this increase in density behind the shock may be attributed to the geometrical convergence or the area contraction of the shock wave. The increase in density is further reinforced by an increase in the value ofθ
, and the decrease in velocity is further reinforced by an increase in the value ofθ
. The behavior of the pressure is more complicated: the pressure profiles behind the shock exhibit non-monotonic variations. The pressure first increases, attains a maximum value and then decreases as we move towards the center of collapse. The pressure increases with an increase in the value ofθ
. For imploding shocks, where the changes in flow variables may be attributed to the geometrical convergence or area contraction of the shock waves, the variations that result from the gas dynamics are evident. The computed values of the similarity exponentδ
are also compared with the Guderley’s [1] result for spherically symmetric case in the Table 1.REFERENCES
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Source of support: Nil, Conflict of interest: None Declared.
